LMIs in Control/pages/Reachable set polytopic

Reachable sets with unit-energy inputs; Polytopic uncertainty

A Reachable set is a set of system States reached under the condition $u=Kx$ . On this page we will look at the problem of finding an controller $K$ , that $E$ contains $RS$ - reachable set.

The System

{\begin{aligned}{\dot {x}}&=Ax+B_{w}w+B_{u}u\\u&=Kx\end{aligned}}

Where:

{\begin{aligned}x&\in R^{n}\\w&\in R^{m}\\u&\in R^{k}\\\end{aligned}}

In case of polytopic uncertainty, we have:

{\begin{aligned}A(t)\;\;B_{w}(t)\;\;B_{u}(t)&\in {\textbf {Co}}\{[A_{1}\;\;B_{w,1}\;\;B_{u;1}],...,[A_{L}\;\;B_{w;L}\;\;B_{u;L}]\}\\\end{aligned}}

Reachable set

The reachable set can be defined:

{\begin{aligned}RS&=\{x(T)|u=Kx;\;\;x(0)=0;\;\;T\geq 0;\;\;\int _{0}^{T}w^{T}wdt<1\}\\\end{aligned}}

The elipsoid $E=\{\varepsilon \in R^{n}|\varepsilon ^{T}Q\varepsilon \leq 1\}$ contains $RS$

The Data

The matrices $A,A_{i}\in R^{n\times n};\;B_{w},B_{w;i}\in R^{n\times m};\;B_{u},B_{u;i}\in R^{n\times k};Q\in R^{n\times n}$ . And

The Optimization Problem

The following optimization problem should be solved:

{\begin{aligned}{\text{Find}}\;&Y>0:\\&QA_{i}^{T}+A_{i}Q+B_{u;i}Y+Y^{T}B_{u;i}^{T}+B_{w;i}B_{w;i}^{T}<0{\text{ for all }}i=1,...n\\&K=YQ^{-1}\end{aligned}}

Conclusion:

This LMI allows us to investigate stability for the robust control problem in the case of polytopic uncertainty and gives on the controller for this case

Implementation:

[1] - Matlab implementation using the YALMIP framework and Mosek solver

Related LMIs:

External Links

A list of references documenting and validating the LMI.

LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes. (3.20.2 page 64)
LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.

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